A patient would like to take a test to determine if he has a nasty disease. Let the variable A denote that the patient has the disease and the variable B denote a positive test,Consider a second test for the same disease. A positive result for this test is denoted by C
If B and C are Conditionally independent on A does it follow that
1) $P(B \cap C )=1 – P(\overline{B}\cap\overline{C})$
2) $p(\overline{B}|\overline{A})=1 -p(B | A )$
3) $p(\overline{C}|\overline{A})=1 – p(C | A ) $
are all three equations above true,
or instead are the following equations the ones that are true
4) $p(\overline{C}|A) = 1 – p(C|A)$
5) $p(\overline{C}|\overline{A}) = 1 – p(C|\overline{A})$
6) $p(\overline{B}|\overline{A}) = 1 – p(B|\overline{A})$
Thank you for any help you can give
Best Answer
I write all this down as an answer because my notes would not fit in a comment.
Given the definition of conditional probability, the de Morgan rule, and assuming that the respective probabilities in the denominators are different from zero then we have the following.
(1) $P(B\cap C)=P\left( \overline{\overline {B\cap C}} \right)=1-P\left(\overline {B\cap C} \right)=1-P\left(\overline B \cup \overline C\right).$ That is, the first statement is false.
(2) $P\left(\overline B|\overline A\right)=\frac{P\left(\overline A \cap\overline B\right)}{P(\overline A)}=\frac{1-P(A \cup B)}{1-P(A)}.$ That is, the second statement is false.
(3) $P\left(\overline C | \overline A\right)=\frac{P\left(\overline C \cap \overline A\right)}{1-P(A)}=\frac{1-P(A\cup C)}{1-P(A)}=$. That is, the third statement is false.
(4) $P\left(\overline C|A\right)=\frac{P\left(A\cap \overline C\right)}{P(A)}=\frac{P(A)-P(A\cap C)}{P(A)}=1-P(C|A)$. That is, the fourth statement is true.
(5) $P\left(\overline C|\overline A\right)=\frac{P\left(\overline C \cap \overline A\right)}{1-P(A)}=\frac{1-P(A \cup C)}{1-P(A)}$. That is, the fifth staement is false.
(6) $P\left(\overline B|\overline A\right)=\frac{P\left(\overline A \cap\overline B\right)}{1-P(A)}=\frac{1-P(A \cup B)}{1-P(A)}$. That is, the sixth statement is false.
All the above have nothing to do with the (at least in this context) undefined assuption of "Conditional independence."